Three – dimensional state of strain and Principal strains in terms of stress
Three – dimensional state of strain : Consider an element subjected to three mutually perpendicular tensile stresses sx , syand sz as shown in the figure below.
If sy and sz were not present the strain in the x direction from the basic definition of Young's modulus of Elasticity E would be equal to
Îx= sx/ E
The effects of sy and sz in x direction are given by the definition of Poisson's ratio ‘ m ' to be equal as -m sy/ E and -m sz/ E
The negative sign indicating that if syand sz are positive i.e. tensile, these they tend to reduce the strain in x direction thus the total linear strain is x direction is given by
Principal strains in terms of stress: In the absence of shear stresses on the faces of the elements let us say that sx , sy , sz are in fact the principal stress. The resulting strain in the three directions would be the principal strains.
i.e. We will have the following relation.
For Two dimensional strain: system, the stress in the third direction becomes zero i.e sz = 0 or s3 = 0
Although we will have a strain in this direction owing to stresses s1& s2 .
Hence the set of equation as described earlier reduces to
Hence a strain can exist without a stress in that direction